Note on quasi-polarized canonical Calabi-Yau threefolds

TL;DR

This paper proves basepoint freeness for multiples of a quasi-polarized canonical Calabi-Yau threefold and explores conditions under which the associated morphism is not birational, with applications to Fano manifolds.

Contribution

It establishes basepoint freeness for multiples of the polarization and characterizes cases where the morphism is not birational, extending to Fano manifolds with specific anticanonical divisors.

Findings

vert mL is basepoint free for m  4.

If vert 4L is not birational and h^0(X,L)  2, then L^3=1.

Classifies Fano manifolds with certain anticanonical divisors, including special hypersurfaces.

Abstract

Let $(X,L)$ be a quasi-polarized canonical Calabi-Yau threefold. In this note, we show that $\vert mL\vert$ is basepoint free for $m\geq 4$. Moreover, if the morphism $\Phi_{\vert 4L\vert}$ is not birational onto its image and $h^0(X,L)\geq 2$, then $L^3=1$. As an application, if $Y$ is a $n$-dimensional Fano manifold such that $-K_Y=(n-3)H$ for some ample divisor $H$, then $\vert mH\vert$ is basepoint free for $m\geq 4$ and if the morphism $\Phi_{\vert 4H\vert}$ is not birational onto its image, then $Y$ is either a weighted hypersurface of degree $10$ in the weighted projective space $\mathbb{P}(1,\cdots,1,2,5)$ or $h^0(Y,H)=n-2$.